A player draws from a bag containing 10 black and 10 white marbles one by one, without putting them back in.
Every time before drawing a marble he guesses the color of the marble he will draw.
He decides to always guess the color that occurs most frequently remaining in the bag (if there is a tie, he chooses either color).

Find the expected number of correct guesses.

Find a formula for the number of correct guesses for starting with X black and Y white marbles.

This problem is a modified version of Guessing suit and addresses an interesting question that came up in the comments.